In dense stellar regions like globular clusters, black hole-neutron star and neutron star-neutron star binaries may form dynamically and merge with significant orbital eccentricities. These mergers are not only a potential gravitational wave source for upcoming detectors like Advanced LIGO, but could also give rise to a number of electromagnetic transients including short gamma-ray bursts. In this talk I will present results from general-relativistic, hydrodynamic simulations of black hole-neutron star and binary neutron star mergers and discuss the large variability with impact parameter that these systems exhibit in dynamics and outcome. I will also discuss ongoing work to model the gravitational wave signals from these events and assess the prospects for detection.